the subprocess amplitudes for the twist-2 and the 2-body twist-3 contributions can easily be
calculated. The familiar twist-2 DA is denoted by \(\phi _\pi \)  whereas \(\phi _{\pi p}\)  and
\(\phi _{\pi \sigma }\)  are the two 2-body twist-3 DAs. The DA \(\phi ^{\prime }_{\pi \sigma }\)  is the derivative of
\(\phi _{\pi \sigma }\)  with respect to the momentum fraction, \(\tau \) , the quark entering the meson carries.
Their definitions can be found for instance in App. A of [1]}. For convenience we omit their
scale dependence (see App. A) in the list of variables in the following. Furthermore, \(f_\pi \) 
in (REF ) is the usual decay constant of the pion (\(f_\pi =132\,\,{\rm MeV}\) ); \(a(f)\)  and \(b(g)\)  represent
flavor (color) labels of the quark and antiquark, respectively. \(N_C\)  is the number of colors.
The Dirac labels are omitted for convenience. The flavor weight factors are
\({\cal C}^{uu}_{\pi ^0}&=&-{\cal C}^{dd}_{\pi ^0}\,=\,\frac{1}{\sqrt{2}}\,, \qquad {\cal C}^{ud}_{\pi ^+}\,=\, {\cal C}^{du}_{\pi ^-}\,=\,1\,.\) 
